[tool] Fix stdin.flush calls on processes started by FakeProcessManager (#151183)

Fixes https://github.com/flutter/flutter/issues/151151

packages/flutter_tools/test/general.shard/fake_process_manager_test.dart

@@ -174,6 +174,13 @@ void main() {
       expect(stderr, 'stderr'.codeUnits);
       expect(stdout, 'stdout'.codeUnits);
     });
+
+    testWithoutContext('stdin should be flushable (all data written is consumed)', () async {
+      final FakeProcess process = FakeProcess();
+      process.stdin.write('hello');
+      // If nothing is listening to the stdin stream, this test will never complete.
+      await process.stdin.flush();
+    });
   });
 
   group(FakeProcessManager, () {

packages/flutter_tools/test/src/fake_process_manager.dart

@@ -160,8 +160,13 @@ class FakeProcess implements io.Process {
          }
          return exitCode;
        }),
-      _stderr = stderr,
-      stdin = stdin ?? IOSink(StreamController<List<int>>().sink),
+        _stderr = stderr,
+        stdin = stdin ??
+            IOSink(
+              StreamController<List<int>>()
+                ..stream.listen((_) {})
+                ..sink,
+            ),
       _stdout = stdout,
       _completer = completer
   {